Navigation

Physics, Topology, Logic and Computation: A Rosetta Stone

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear
that underlying these diagrams is a powerful analogy between quantum physics and topology. Namely, a linear
operator behaves very much like a `cobordism': a manifold representing spacetime, going between two
manifolds representing space. This led to a burst of work on topological quantum field theory and `quantum
topology'. But this was just the beginning: similar diagrams can be used to reason about logic, where
they represent proofs, and computation, where they represent programs. With the rise of interest in quantum
cryptography and quantum computation, it became clear that there is extensive network of analogies between
physics, topology, logic and computation. In this expository paper, we make some of these analogies precise
using the concept of `closed symmetric monoidal category'. We assume no prior knowledge of category
theory, proof theory or computer science.

I am not sure whether this should be categorized as "Fun" instead of "Theory", given that "We assume no prior knowledge of category theory, proof theory or computer science".

At least one pair from the title (logic and computation) should ring some bells...

This is probably redundant, but I should point out that this paper (in its various draft versions) has been discussed at length on the n-Category cafe. Here's what seems to be the latest thread, which also links an interesting paper by Peter Selinger which surveys graphical notations for monoidal categories, with obvious implications for visual programming languages. Also, another paper by B. Coecke and E. O. Paquette, which takes a different angle on the same topic.